Group #22561456 — subgroups

Subgroups — #22561456

Order 8576.

298 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14524043	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523968	C₂	1	2	yes	maximal
3	C₂×#14523968	C₂	1	2	yes	maximal
4	#16863294	C₂	1	1	yes	maximal, characteristic
5	#336×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
7	#14524043	—	2	4	no	—
8	C₂×C₄⋊Dic₆₇	—	2	1	no	—
9	C₂×C₄⋊Dic₆₇	—	2	1	no	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₄	—	2	1	no	—
13	#14523968	—	2	4	no	—
14	#14523968	—	2	4	no	—
15	C₆₇⋊#95	—	2	1	no	—
16	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₃₄	—	2	2	no	—
19	C₂×Dic₁₃₄	—	2	2	no	—
20	C₂×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₁₃₄	—	2	2	no	—
22	C₂×Dic₁₃₄	—	2	2	no	—
23	C₂×Dic₁₃₄	D₄	1	4	yes	—
24	C₂²×Dic₆₇	—	2	1	no	—
25	C₂²×Dic₆₇	—	2	2	no	—
26	C₂²×Dic₆₇	—	2	1	no	—
27	C₄⋊Dic₆₇	—	2	4	no	—
28	C₄⋊Dic₆₇	—	2	4	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	2	no	—
31	C₄⋊Dic₆₇	D₄	1	2	yes	—
32	Dic₆₇⋊C₄	—	4	2	no	—
33	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₆₇	—	2	2	no	—
37	C₂×C₈×C₆₇	—	2	2	no	—
38	C₂×C₈×C₆₇	—	2	2	no	—
39	C₄²×C₆₇	—	2	2	no	—
40	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	2	1	no	—
42	C₂×C₄×C₆₇	—	2	1	no	—
43	C₂×C₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
46	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₇	#41	1	2	yes	—
50	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
51	C₂×Dic₆₇	—	2	4	no	—
52	C₂×Dic₆₇	—	2	2	no	—
53	C₂×Dic₆₇	—	4	2	no	—
54	C₂×Dic₆₇	—	4	2	no	—
55	C₂×Dic₆₇	—	4	1	no	—
56	C₂×Dic₆₇	—	4	1	no	—
57	C₂×Dic₆₇	—	4	1	no	—
58	C₂×Dic₆₇	—	4	1	no	—
59	C₂×Dic₆₇	—	4	1	no	—
60	Dic₁₃₄	—	2	4	no	—
61	Dic₁₃₄	—	2	4	no	—
62	Dic₁₃₄	—	4	2	no	—
63	Dic₁₃₄	—	4	2	no	—
64	C₈×C₆₇	—	4	2	no	—
65	C₂²×C₆₇	#120	1	1	yes	characteristic
66	C₂²×C₆₇	#121	1	1	yes	characteristic
67	C₂²×C₆₇	#124	1	1	yes	characteristic
68	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₆₇	#137	1	1	yes	characteristic
71	C₂²×C₆₇	#137	1	1	yes	characteristic
72	C₄×C₆₇	—	2	2	no	—
73	C₄×C₆₇	—	2	2	no	—
74	C₄×C₆₇	#121	1	2	yes	—
75	C₄×C₆₇	#123	1	2	yes	—
76	Dic₆₇	—	4	4	no	—
77	Dic₆₇	—	4	2	no	—
78	Dic₆₇	—	8	1	no	—
79	C₂×C₆₇	#394	1	1	yes	characteristic
80	C₂×C₆₇	#448	1	1	yes	characteristic
81	C₂×C₆₇	#451	1	1	yes	characteristic
82	C₂×C₆₇	#461	1	1	yes	characteristic
83	C₂×C₆₇	#462	1	1	yes	characteristic
84	C₂×C₆₇	#487	1	1	yes	characteristic
85	C₂×C₆₇	#491	1	1	yes	characteristic
86	#2043	—	67	1	no	maximal, Sylow
87	C₆₇	#2043	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊Q₈	—	67	1	no	—
89	C₂×#103	—	67	2	no	—
90	C₂×#103	—	67	2	no	—
91	#389	—	67	1	no	—
92	#336	D₆₇	1	1	yes	characteristic
93	C₂²×Q₈	—	67	2	no	—
94	C₂²×C₈	C₂×D₆₇	1	2	yes	—
95	C₄⋊Q₈	—	134	4	no	—
96	C₂×C₄⋊C₄	—	67	1	no	—
97	C₂×C₄⋊C₄	—	67	1	no	—
98	C₂×C₄⋊C₄	—	134	1	no	—
99	C₂×C₄⋊C₄	—	134	1	no	—
100	C₂×C₄⋊C₄	—	134	1	no	—
101	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
102	#103	—	134	4	no	—
103	#103	—	134	4	no	—
104	#95	—	134	1	no	—
105	C₂×Q₈	—	67	4	no	—
106	C₂×Q₈	—	134	2	no	—
107	C₂×Q₈	—	134	2	no	—
108	C₂×Q₈	—	134	2	no	—
109	C₂×Q₈	—	134	2	no	—
110	C₂×Q₈	—	134	2	no	—
111	C₂²×C₄	—	134	1	no	—
112	C₂²×C₄	—	134	1	no	—
113	C₂²×C₄	—	134	2	no	—
114	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
115	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₄⋊C₄	—	268	2	no	—
121	C₄⋊C₄	—	67	2	no	—
122	C₄⋊C₄	—	134	4	no	—
123	C₄⋊C₄	—	134	4	no	—
124	C₄⋊C₄	—	134	2	no	—
125	C₄⋊C₄	—	134	1	no	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
128	Q₈	—	268	2	no	—
129	Q₈	—	268	2	no	—
130	Q₈	—	134	4	no	—
131	Q₈	—	134	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	268	2	no	—
137	C₂×C₄	—	268	2	no	—
138	C₂×C₄	—	268	1	no	—
139	C₂×C₄	—	268	1	no	—
140	C₂×C₄	—	268	1	no	—
141	C₂×C₄	—	268	1	no	—
142	C₂×C₄	—	268	1	no	—
143	C₂×C₄	—	134	4	no	—
144	C₂×C₄	—	134	2	no	—
145	C₂×C₄	#11761617	1	2	yes	—
146	C₂×C₄	Dic₂₆₈	1	2	yes	—
147	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
148	C₂×C₄	D₄×D₆₇	1	2	yes	—
149	C₂×C₄	#11761628	1	2	yes	—
150	C₂×C₄	#11761629	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	#14524013	1	1	yes	characteristic
153	C₂²	#14524018	1	1	yes	characteristic
154	C₂²	#14524031	1	1	yes	characteristic
155	C₂²	C₂×#11761617	1	1	yes	characteristic
156	C₂²	C₂×Dic₂₆₈	1	1	yes	characteristic
157	C₂²	#14524076	1	1	yes	characteristic
158	C₂²	#14524076	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	268	4	no	—
162	C₄	—	268	2	no	—
163	C₄	—	536	1	no	—
164	C₄	#14524016	1	2	yes	—
165	C₄	#14524028	1	2	yes	—
166	C₂	#16863314	1	1	yes	characteristic
167	C₂	#16863537	1	1	yes	characteristic
168	C₂	#16863554	1	1	yes	characteristic
169	C₂	#16863602	1	1	yes	characteristic
170	C₂	#16863607	1	1	yes	characteristic
171	C₂	#16863747	1	1	yes	characteristic
172	C₂	#16863765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic

C₂×#14523968

C₂

yes

maximal

C₂×#14523968

C₂

yes

maximal

#16863294

C₂

yes

maximal, characteristic

#336×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

#14524043

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

—

#14523968

—

#14523968

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

Q₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₂×SD₁₆

yes

characteristic

C₂²×C₆₇

C₂×Q₁₆

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

—

C₄×C₆₇

#123

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₆₇

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₆₇

yes

—

C₄⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

100

C₂×C₄⋊C₄

—

134

—

101

C₂×C₄²

C₂×D₆₇

yes

characteristic

102

#103

—

134

—

103

#103

—

134

—

104

#95

—

134

—

105

C₂×Q₈

—

106

C₂×Q₈

—

134

—

107

C₂×Q₈

—

134

—

108

C₂×Q₈

—

134

—

109

C₂×Q₈

—

134

—

110

C₂×Q₈

—

134

—

111

C₂²×C₄

—

134

—

112

C₂²×C₄

—

134

—

113

C₂²×C₄

—

134

—

114

C₂²×C₄

D₂₆₈

yes

characteristic

115

C₂²×C₄

D₂₆₈

yes

characteristic

116

C₂²×C₄

C₂²×D₆₇

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

268

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

134

—

123

C₄⋊C₄

—

134

—

124

C₄⋊C₄

—

134

—

125

C₄⋊C₄

—

134

—

126

C₄²

—

127

C₂³

C₂×D₂₆₈

yes

characteristic

128

Q₈

—

268

—

129

Q₈

—

268

—

130

Q₈

—

134

—

131

Q₈

—

134

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

268

—

137

C₂×C₄

—

268

—

138

C₂×C₄

—

268

—

139

C₂×C₄

—

268

—

140

C₂×C₄

—

268

—

141

C₂×C₄

—

268

—

142

C₂×C₄

—

268

—

143

C₂×C₄

—

134

—

144

C₂×C₄

—

134

—

145

C₂×C₄

#11761617

yes

—

146

C₂×C₄

Dic₂₆₈

yes

—

147

C₂×C₄

D₄×D₆₇

yes

characteristic

148

C₂×C₄

D₄×D₆₇

yes

—

149

C₂×C₄

#11761628

yes

—

150

C₂×C₄

#11761629

yes

characteristic

151

C₈

—

152

C₂²

#14524013

yes

characteristic

153

C₂²

#14524018

yes

characteristic

154

C₂²

#14524031

yes

characteristic

155

C₂²

C₂×#11761617

yes

characteristic

156

C₂²

C₂×Dic₂₆₈

yes

characteristic

157

C₂²

#14524076

yes

characteristic

158

C₂²

#14524076

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

268

—

162

C₄

—

268

—

163

C₄

—

536

—

164

C₄

#14524016

yes

—

165

C₄

#14524028

yes

—

166

C₂

#16863314

yes

characteristic

167

C₂

#16863537

yes

characteristic

168

C₂

#16863554

yes

characteristic

169

C₂

#16863602

yes

characteristic

170

C₂

#16863607

yes

characteristic

171

C₂

#16863747

yes

characteristic

172

C₂

#16863765

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#336×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#336	D₆₇
hypocenter	C₆₇	#2043
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#336×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
87. C₆₇	#2043	semidirect

H ≅ G/N

type

87. C₆₇

#2043

semidirect

Subgroups — #22561456

Conjugacy classes of subgroups

Complements